Graphing The Rational Function Y=x/(2x+2) Identifying The Untouched Quadrant

by THE IDEN 77 views

In the realm of mathematics, rational functions hold a significant position, offering a unique blend of algebraic expressions and graphical representations. These functions, expressed as the ratio of two polynomials, often exhibit fascinating behaviors, including asymptotes, intercepts, and distinct branches. This article delves into the intricacies of graphing the rational function y = x / (2x + 2), aiming to provide a comprehensive understanding of its characteristics and graphical representation. We will explore the key steps involved in graphing rational functions, including identifying asymptotes, intercepts, and analyzing the function's behavior in different intervals. Furthermore, we will pinpoint the specific quadrant that neither branch of the function traverses, providing a clear and concise answer to the posed question.

Understanding Rational Functions

At its core, a rational function is simply a function that can be expressed as the quotient of two polynomials. In mathematical notation, this is represented as f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions. The domain of a rational function is all real numbers except for the values of x that make the denominator Q(x) equal to zero, as division by zero is undefined. These values are crucial in determining the vertical asymptotes of the function.

Rational functions are characterized by several key features that influence their graphs. Asymptotes, both vertical and horizontal, play a pivotal role in defining the function's behavior as x approaches certain values or infinity. Intercepts, where the graph crosses the x-axis and y-axis, provide additional anchor points for plotting the function. The overall shape of the graph is further influenced by the degree of the polynomials in the numerator and denominator, as well as the presence of any common factors.

Step-by-Step Graphing of y=x/(2x+2)

To effectively graph the rational function y = x / (2x + 2), we will follow a systematic approach, breaking down the process into manageable steps. This methodical approach will ensure that we capture all the essential features of the function and accurately represent it on a graph.

1. Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the rational function equals zero. To find these, we set the denominator 2x + 2 equal to zero and solve for x:

2x + 2 = 0
2x = -2
x = -1

Thus, the vertical asymptote is at x = -1. This means the function will approach infinity (or negative infinity) as x gets closer to -1 from either side.

2. Identify Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, we compare the degrees of the numerator and denominator polynomials.

In our case, the numerator x has a degree of 1, and the denominator 2x + 2 also has a degree of 1. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 2. Therefore, the horizontal asymptote is y = 1/2.

3. Find Intercepts

Intercepts are the points where the graph crosses the x-axis and y-axis. To find the x-intercept, we set y = 0 and solve for x:

0 = x / (2x + 2)
0 = x

So, the x-intercept is at (0, 0).

To find the y-intercept, we set x = 0 and solve for y:

y = 0 / (2(0) + 2)
y = 0 / 2
y = 0

The y-intercept is also at (0, 0).

4. Analyze the Function's Behavior

To fully understand the function's behavior, we need to analyze how it behaves in the intervals defined by the vertical asymptote. Our vertical asymptote is at x = -1, which divides the x-axis into two intervals: x < -1 and x > -1. We will choose test points in each interval to determine the sign of the function.

  • Interval x < -1: Let's choose x = -2. Then, y = (-2) / (2(-2) + 2) = -2 / (-2) = 1. So, in this interval, the function is positive.
  • Interval x > -1: Let's choose x = 0 (we already know this point, but it confirms our intercept). Then, y = 0 / (2(0) + 2) = 0. Let's choose x = 1. Then, y = 1 / (2(1) + 2) = 1 / 4. So, in this interval, the function is positive for x > 0.

5. Sketch the Graph

Now that we have all the necessary information, we can sketch the graph. We know the vertical asymptote is at x = -1, the horizontal asymptote is at y = 1/2, and the function passes through the origin (0, 0). We also know the function is positive for x < -1 and positive for x > 0. As x approaches -1 from the left, the function approaches positive infinity, and as x approaches -1 from the right, the function approaches negative infinity. Based on this information, we can sketch two distinct branches of the hyperbola.

Identifying the Untouched Quadrant

Now, let's determine which quadrant neither branch of the rational function y = x / (2x + 2) passes through. Recall that the coordinate plane is divided into four quadrants:

  • Quadrant I: x > 0, y > 0
  • Quadrant II: x < 0, y > 0
  • Quadrant III: x < 0, y < 0
  • Quadrant IV: x > 0, y < 0

Based on our analysis and the sketch of the graph, we can observe the following:

  • The left branch of the function (where x < -1) lies in Quadrant II (x is negative, y is positive) as it approaches the horizontal asymptote y = 1/2 from above and goes up toward positive infinity as it approaches the vertical asymptote.
  • The right branch of the function lies in Quadrant I (for x > 0, both x and y are positive) and Quadrant IV (for -1< x <0, x is positive and y is negative). As x increases from 0, the y value approaches the horizontal asymptote, 1/2, staying positive. As x approaches -1 from the right, the y value approaches negative infinity.

Therefore, neither branch of the rational function y = x / (2x + 2) passes through Quadrant III (x < 0, y < 0). This is because the function is positive for x < -1 (Quadrant II) and approaches negative infinity as x approaches -1 from the right (Quadrant IV).

Conclusion: Unveiling the Graphical Behavior of Rational Functions

In conclusion, by meticulously analyzing the rational function y = x / (2x + 2), we have successfully graphed the function and identified that Quadrant III is the quadrant that neither branch of the function passes through. This process involved several crucial steps, including identifying vertical and horizontal asymptotes, finding intercepts, analyzing the function's behavior in different intervals, and sketching the graph. Understanding these steps is crucial for effectively graphing any rational function.

The exploration of rational functions provides a fascinating glimpse into the world of mathematical relationships and their graphical representations. By mastering the techniques outlined in this article, students and enthusiasts alike can confidently navigate the intricacies of graphing rational functions and gain a deeper appreciation for the elegance and power of mathematics.

Therefore, the answer is (C).